# How do you evaluate the integral int e^(-absx) from -oo to oo?

Jul 7, 2016

2

#### Explanation:

graph{e^(- | x|) [-10, 10, -5, 5]}

use the symmetry so that it becomes

$\textcolor{red}{2 \times} {\int}_{0}^{\infty} \mathrm{dx} q \quad {e}^{- x}$

ie we are integrating in the region $x \ge 0$ using the fact that $| x | = x$

$= 2 {\left[- {e}^{- x}\right]}_{0}^{\infty}$

$= 2 {\left[{e}^{- x}\right]}_{\infty}^{0}$

2

to test for the symmetry use the even funcition test ie does $f \left(- x\right) = f \left(x\right)$

here

$f \left(- x\right) = {e}^{- \left\mid - x \right\mid} = {e}^{- \left\mid x \right\mid} = f \left(x\right)$